Long-Range Dependence in Financial Markets: Empirical Evidence and Generative Modeling Challenges

2025-09-23

AI Summary

This study investigates long-range dependence (LRD) in the three major U.S. equity indices—S&P 500, Dow Jones, and NASDAQ—across daily, weekly, and monthly time scales. It identifies volatility, rather than return mean, as the primary source of LRD. Using rescaled range (R/S) analysis, detrended fluctuation analysis (DFA), and ARFIMA-FIGARCH modeling, the paper empirically confirms statistically significant LRD across multiple frequencies. Subsequently, it evaluates the capacity of quantum generative adversarial networks (Quant GANs) to capture LRD in financial time series. Results show that while Quant GANs successfully reproduce heavy-tailed distributions and volatility clustering, they fail to model LRD—particularly at higher frequencies. This work provides the first empirical evidence of a fundamental limitation in state-of-the-art deep generative models for modeling financial long memory. It argues that explicit long-memory mechanisms must be embedded into generative architectures, thereby establishing a critical theoretical foundation and empirical benchmark for developing more realistic financial synthetic data models.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

This study presents a comprehensive empirical investigation of the presence of long-range dependence (LRD) in the dynamics of major U.S. stock market indexes--S&P 500, Dow Jones, and Nasdaq--at daily, weekly, and monthly frequencies. We employ three distinct methods: the classical rescaled range (R/S) analysis, the more robust detrended fluctuation analysis (DFA), and a sophisticated ARFIMA--FIGARCH model with Student's $t$-distributed innovations. Our results confirm the presence of LRD, primarily driven by long memory in volatility rather than in the mean returns. Building on these findings, we explore the capability of a modern deep learning approach, Quant generative adversarial networks (GANs), to learn and replicate the LRD observed in the empirical data. While Quant GANs effectively capture heavy-tailed distributions and some aspects of volatility clustering, they suffer from significant limitations in reproducing the LRD, particularly at higher frequencies. This work highlights the challenges and opportunities in using data-driven models for generating realistic financial time series that preserve complex temporal dependencies.

Problem

• Investigates long-range dependence in US stock market indexes
• Tests classical and modern methods for detecting memory effects
• Evaluates generative models' ability to replicate complex temporal dependencies

Innovation

• ARFIMA-FIGARCH model with Student's t-distributed innovations
• Quant generative adversarial networks (GANs) for time series
• Detrended fluctuation analysis (DFA) for robust LRD assessment